Résume  By a result of MazzeoPacard, every hyperbolic quasiFuchsian manifold admits a compact subset whose complement is foliated by constant mean curvature (CMC) surfaces. However, there exist quasiFuchsian manifolds that do not admit a global CMC foliation. A conjecture due to Thurston asserts that every almostFuchsian manifold has such a global CMC foliation. In this talk I will discuss a partial result in this direction, obtained in a joint work with Diptaishik Choudhury and Filippo Mazzoli: every quasiFuchsian manifold in a neighbourhood of the Fuchsian locus is (uniquely and monotonically) foliated by CMC surfaces. Time permitting, in the final part of the talk I will explain how these foliations induce Hamiltonian flows on the cotangent bundle of Teichmüller space.
